On family of potential wells and applications to semilinear hyperbolic equations and parabolic equations. (September 2020) - Chen, Tianlong; Chen, Yuxuan
Abstract: We study the initial boundary value problem for semilinear hyperbolic equations and parabolic equations with convex function nonlinear terms. By introducing a family of potential wells we not only obtain the invariant sets and vacuum isolation of solutions, but also give a threshold result of global existence and nonexistence of solutions. Moreover we discuss the global existence of solution for problem with critical initial condition E ( 0 ) = d (or J ( u 0 ) = d ), which fills some important gaps regarding this problem. Finally we prove the global nonexistence of solution and asymptotic behavior of solution for semilinear parabolic equations.